Maps preserving the Douglas solution of operator equations

Zsigmond Tarcsay

doi:10.1016/j.jmaa.2021.125802

Abstract

We consider bijective maps ϕ on the full operator algebra B(H) of an infinite dimensional Hilbert space with the property that, for every A,B,X∈B(H), X is the Douglas solution of the equation A=BX if and only if Y=ϕ(X) is the Douglas solution of the equation ϕ(A)=ϕ(B)Y. We prove that those maps are implemented by a unitary or anti-unitary map U, i.e., ϕ(A)=UAU⁎.

Highlights

Operator equations of the form (1.1)BX = A arise in many problems in engineering, physics, and statistics
In this paper we consider the following nonlinear preserver problem: Let φ : B(H) → B(H) be a bijective map with the property that, for every triple A, B, X of bounded operators in B(H), X is the Douglas solution of the equation A = BX if and only if Y = φ(X) is the Douglas solution of the equation φ(A) = φ(B)Y . (Shortly, we say in that case that φ preserves the Douglas solution in both directions.) In this note we describe the form all such transformation φ
Our result shows that the structure of those mappings is quite rigid, namely, every Douglas solution preserving map φ is of the form φ(A) = U AU ∗, A ∈ B(H), for a fixed unitary or anti-unitary map U

Summary

Introduction

G. Douglas considered the problem in the context of Hilbert space operators. In this paper we consider the following nonlinear preserver problem: Let φ : B(H) → B(H) be a bijective map with the property that, for every triple A, B, X of bounded operators in B(H), X is the Douglas solution of the equation A = BX if and only if Y = φ(X) is the Douglas solution of the equation φ(A) = φ(B)Y .

Results

Conclusion